What is the exact definition of polynomial functions? I'm trying to understand the difference between polynomial functions and the evaluation homomorphisms. I noticed that I don't know what's the exact definition of a polynomial function, although I've used it since I was in school.
I'm confused.
Thanks in advance
 A: *

*A polynomial with coefficients in a commutative ring $R$ is a formal sum $r_n x^n + \ldots + r_1 x + r_0$. It is not a function. It is merely a finite sum of terms of the form $r_i x^i$, where $x$ is some abstract symbol. The ring structure of $R$ lets us make these symbols into a ring, that we denote $R[x]$. 

*For each $r \in R$, there is a ring homomorphism $e_r:R[x] \rightarrow R$. These are the evaluation maps, defined by replacing each instance of $x$ in the formal sum of a polynomial with $r$ and considering this as an element of $R$. We sometimes denote $e_r(f(x))$ as $f(r)$, but this is not the same as the formal polynomial $f(x)$. $f(r)$ for $r \in R$ is actually an element of the ring, while the formal polynomial $f(x)$ is not.

*For each polynomial $f(x) \in R[x]$, there is a map $R \rightarrow R$ defined by $r \mapsto e_r(f(x)) = f(r)$. This is not a ring homomorphism! (For instance, the polynomial function on $\mathbb{R}$ defined by $f(x) = x^2$ is not a homomorphism of rings.) It is a function determined by a polynomial, and so we call such a function a polynomial function.
All of these things can be generalized to polynomials in multiple variables, as well.
A note: Why the fuss about formal polynomials and the functions they define? Typically, you're safe not making a distinction between the formal sum and the function it defines. However, some issues can arise and you need to be aware of them. For instance, consider the polynomial $x^2 - x = (x-1)x$ with coefficients in the ring $\mathbb{F}_2$, the field with two elements. This polynomial, formally, is not zero... it has nonzero coefficients! However, the function it defines on $\mathbb{F}_2$ is the zero function, since there are only two points in $\mathbb{F}_2$ and the evaluation homomorphism for each point sends the function to zero.
